If $n$ and $m$ are both even, then the tricks of the previous slide do not help. Instead, we can either integrate by parts (using the "go in a circle" trick) or use double-angle formulas. Most people find the double-angle formulas to be easier, and that's what this slide's video is about.

Since
$$
\cos(2x)=\cos^2(x)-\sin^2(x) $$ $$\qquad= 1-2\sin^2(x)$$ $$\qquad= 2\cos^2(x)-1,
$$
we can rewrite products of $\sin^2(x)$ and $\cos^2(x)$ in terms of $\cos(2x)$:$$\sin^2(x) = \frac{1-\cos(2x)}{2}$$ $$\cos^2(x) = \frac{1+\cos(2x)}{2}.$$This converts $\sin^n(x)\cos^m(x)$, with $n$ and $m$ even, into a polynomial in $\cos(2x)$ of degree $(n+m)/2$. The odd powers of $\cos(2x)$ can be handled as in the previous slide. To integrate the even powers, apply the double-angle trick again, getting a polynomial in $\cos(4x)$. Repeat as many times as necessary.